L(s) = 1 | + (−0.779 − 0.626i)2-s + (0.568 + 0.822i)3-s + (0.215 + 0.976i)4-s + (0.568 + 0.822i)5-s + (0.0724 − 0.997i)6-s + (0.0724 + 0.997i)7-s + (0.443 − 0.896i)8-s + (−0.354 + 0.935i)9-s + (0.0724 − 0.997i)10-s + (−0.681 + 0.732i)12-s + (−0.644 − 0.764i)13-s + (0.568 − 0.822i)14-s + (−0.354 + 0.935i)15-s + (−0.906 + 0.421i)16-s + (−0.981 − 0.192i)17-s + (0.861 − 0.506i)18-s + ⋯ |
L(s) = 1 | + (−0.779 − 0.626i)2-s + (0.568 + 0.822i)3-s + (0.215 + 0.976i)4-s + (0.568 + 0.822i)5-s + (0.0724 − 0.997i)6-s + (0.0724 + 0.997i)7-s + (0.443 − 0.896i)8-s + (−0.354 + 0.935i)9-s + (0.0724 − 0.997i)10-s + (−0.681 + 0.732i)12-s + (−0.644 − 0.764i)13-s + (0.568 − 0.822i)14-s + (−0.354 + 0.935i)15-s + (−0.906 + 0.421i)16-s + (−0.981 − 0.192i)17-s + (0.861 − 0.506i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0949 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0949 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.07807030516 + 0.08587418843i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.07807030516 + 0.08587418843i\) |
\(L(1)\) |
\(\approx\) |
\(0.7148820629 + 0.2866846164i\) |
\(L(1)\) |
\(\approx\) |
\(0.7148820629 + 0.2866846164i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 131 | \( 1 \) |
good | 2 | \( 1 + (-0.779 - 0.626i)T \) |
| 3 | \( 1 + (0.568 + 0.822i)T \) |
| 5 | \( 1 + (0.568 + 0.822i)T \) |
| 7 | \( 1 + (0.0724 + 0.997i)T \) |
| 13 | \( 1 + (-0.644 - 0.764i)T \) |
| 17 | \( 1 + (-0.981 - 0.192i)T \) |
| 19 | \( 1 + (-0.981 + 0.192i)T \) |
| 23 | \( 1 + (-0.168 + 0.985i)T \) |
| 29 | \( 1 + (-0.779 + 0.626i)T \) |
| 31 | \( 1 + (0.885 - 0.464i)T \) |
| 37 | \( 1 + (0.399 - 0.916i)T \) |
| 41 | \( 1 + (0.681 - 0.732i)T \) |
| 43 | \( 1 + (-0.0241 + 0.999i)T \) |
| 47 | \( 1 + (-0.998 - 0.0483i)T \) |
| 53 | \( 1 + (0.309 - 0.951i)T \) |
| 59 | \( 1 + (0.485 - 0.873i)T \) |
| 61 | \( 1 + (-0.309 + 0.951i)T \) |
| 67 | \( 1 + (0.0241 + 0.999i)T \) |
| 71 | \( 1 + (-0.989 - 0.144i)T \) |
| 73 | \( 1 + (0.809 + 0.587i)T \) |
| 79 | \( 1 + (0.943 + 0.331i)T \) |
| 83 | \( 1 + (0.748 - 0.663i)T \) |
| 89 | \( 1 + (-0.809 - 0.587i)T \) |
| 97 | \( 1 + (-0.970 - 0.239i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−19.69604882386421121910542397612, −19.39385633649768846639664520152, −18.34366247086943203628187126069, −17.66897386445449556171585202652, −16.925444127361035277855993930898, −16.67569988846544926488626634057, −15.41439336688094316050894986286, −14.65303321260820923643921369791, −13.85531517651621156319342793990, −13.38833069356638108244621496967, −12.50106836462158855703979941520, −11.47711461156851772379247591166, −10.477611256438663503227479240181, −9.64918600062467105913527279562, −8.94622407450702236517625917030, −8.284508710298367315518976999429, −7.559765694061724780405750506302, −6.53377303182101635828195137022, −6.32803871664543127104169080333, −4.84800249610910508602160931837, −4.211607530840454265851458986154, −2.45081208043765864976455974043, −1.81275782708689450510769964297, −0.87281867017476878045586061501, −0.02831692593506217690379861393,
1.89075762550674800316285773796, 2.430688184142193706684118625310, 3.10168313783637120651635045042, 4.05646626425125812101675143256, 5.18822436380586279851807070275, 6.15219903079068372954952315786, 7.27924124288433569925030653614, 8.11049116058796992070317434039, 8.91536194839133171150582689169, 9.57948440402342127198583888933, 10.14036640114957878669378177273, 11.01439854246268490411919748402, 11.48899388577264948072634263142, 12.735791804312925079338165416780, 13.33407923863990820035243547469, 14.47805363712786365872529314360, 15.10807582581216615478824010533, 15.73092466220827181897543088825, 16.64858170287275109821968531466, 17.71014848705916679688871738740, 17.907810760090058074409503977955, 19.15128692760087165757784541139, 19.37332457849755484455720614174, 20.33805577285133877634997978700, 21.20745005286488518663401649285